Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the match pair procedure
Résumé
In this work we investigate the notion of action or coaction of a finite quantum groupoid in von Neumann algebras context. In particular we prove a double crossed product theorem and prove the existence of an universal von Neumann algebra on which any finite groupoids acts outerly. In previous works, N. Andruskiewitsch and S.Natale define for any match pair of groupoids two $C^*$-quantum groupoids in duality, we give here an interpretation of them in terms of crossed products of groupoids using a multiplicative partial isometry which gives a complete description of these structures. In a next work we shall give a third description of these structures dealing with inclusions of depth two inclusions of von Neumann algebras associated with outer actions of match pairs of groupoids, and a study, in the same spirit, of an other extension of the match pair procedure.